Anisotropic partial regularity criteria for the Navier-Stokes - PowerPoint PPT Presentation

Anisotropic partial regularity criteria for the Navier-Stokes equations Walter Rusin Department of Mathematics Mathflows 2015 Porquerolles September 17, 2015

The question of regularity of the weak solutions of the 3D incompressible Navier-Stokes equations 3 � ∂ t u − ∆ u + ∂ j ( u j u ) + ∇ p = 0 (NSE) j =1 div u = 0 where u ( x , t ) = ( u 1 ( x , t ) , u 2 ( x , t ) , u 3 ( x , t )) and p ( x , t ) denote the unknown velocity and the pressure, has remained open since 1930’s (works of Leray ’34). 2 of 35

For initial data u 0 ( x ) ∈ L 2 ( R 3 ), the energy estimate � t � u ( · , t ) � 2 �∇ u ( · , s ) � 2 L 2 ds ≤ � u 0 � 2 L 2 + 2 L 2 0 plays a crucial role in establishing the existence of global weak solutions. In particular, for any T > 0 we obtain u ∈ L ∞ ((0 , T ) , L 2 ( R 3 )) ∩ L 2 ((0 , T ) , ˙ H 1 ( R 3 )) . 3 of 35

The Sobolev embedding and interpolation yield that for such u ( x , t ) we have u ∈ L 10 / 3 t , x ((0 , T ) × R 3 ) or any other Lebesgue space L p t L q x ((0 , T ) × R 3 ) provided that p + 3 2 q = 3 2 . 4 of 35

The available regularity criteria (e.g. Ladyzhenskaya-Prodi-Serrin) require however p + 3 2 q ≤ 1 . Note that not only we cannot draw any conclusion but there is also a substantial gap between what is at hand and what would be necessary to conclude regularity. 5 of 35

The theory of partial regularity for the NSE aims at estimating the Husdorff dimension of the (possible) singular set and development of interior regularity criteria. Recall that a point is regular if there exists a neighborhood in which u is bounded (and thus H¨ older continuous); otherwise, the point is called singular . 6 of 35

The program has been started by Scheffer in the 70’s. In a classical paper, Caffarelli, Kohn, and Nirenberg proved that the one-dimensional parabolic Hausdorff measure (parabolic Hausdorff length) of the singular set equals zero. In this case, the interior regularity criterion is as follows: There exist two constants ǫ CKN ∈ (0 , 1] and α ∈ (0 , 1) such that if � ( | u | 3 + | p | 3 / 2 ) dxdt ≤ ǫ CKN Q 1 then � u ( x , t ) � C α ( Q 1 / 2 ) < ∞ where Q r = { ( x , t ) : | x | < r , − r 2 ≤ t ≤ 0 } . 7 of 35

Alternative proofs were given by Ladyzhenskaya and Ser¨ egin, Kukavica, Lin, Vasseur, and Wolf. The problem of partial regularity of the solutions of the Navier–Stokes equations has been since then addressed in various contexts and a variety of interior regularity criteria has been proposed. 8 of 35

In particular Wolf proved the following: There exists ǫ W > 0 such that if � | u | 3 dxdt ≤ ǫ W Q 1 then the solution u ( x , t ) is regular at the point (0 , 0). 9 of 35

Fairly recently, Wang and Zhang proved an anisotropic interior regularity criterion, which states: For every M > 0 there exists ǫ WZ ( M ) > 0 such that if � ( | u | 3 + | p | 3 / 2 ) dxdt ≤ M Q 1 and � | u h | 3 dxdt ≤ ǫ WZ ( M ) Q 1 where u h = ( u 1 , u 2 ), then the solution u ( x , t ) is regular at the point (0 , 0). 10 of 35

One component regularity/anisotropic results: • Cao, Titi • Chemin, Zhang • Zhou • Raugel, Sell • He • Mucha • Nestupa, Novotny, Penel, Pokorny • Kukavica, Ziane, R. • ... 11 of 35

The purpose of this talk is to present interior regularity criteria involving only one component of the velocity. We prove, with a different argument, the following statement: For every M > 0 there exists a constant ǫ ( M ) > 0 such that if � ( | u | 3 + | p | 3 / 2 ) dxdt ≤ M Q 1 and � | u 3 | 3 ≤ ǫ ( M ) Q 1 then u ( x , t ) is regular at the point (0 , 0). 12 of 35

Let D be an open, bounded, and connected subset of R 3 × (0 , ∞ ). We work in the class of suitable weak solutions, that is ( u , p ) satisfies t L 2 x ( D ) ∩ L 2 t H 1 x ( D ) and p ∈ L 3 / 2 ( D ), (i) u ∈ L ∞ (ii) the Navier-Stokes equations (NSE) are satisfied in the weak sense (iii) the local energy inequality holds in D , i.e., � � � � R 3 | u | 2 φ dx |∇ u | 2 φ dxdt � + 2 � � R 3 × ( −∞ , T ] T � � � � | u | 2 ( ∂ t φ + ∆ φ ) + ( | u | 2 + 2 p ) u · ∇ φ ≤ dxdt R 3 × ( −∞ , T ] for all φ ∈ C ∞ 0 ( D ) such that φ ≥ 0 in D and almost all T ∈ R . 13 of 35

Theorem 1 Let ( u , p ) be a suitable weak solution of (NSE) in a neighborhood of Q r ( x 0 , t 0 ) ⊂ D which satisfies 1 � | u | 3 + | p | 3 / 2 � � dxdt ≤ M . (1) r 2 Q r ( x 0 , t 0 ) Then there exists ǫ > 0 depending on M such that if 1 � | u 3 | 3 dxdt ≤ ǫ (2) r 2 Q r ( x 0 , t 0 ) then u is regular at ( x 0 , t 0 ). 14 of 35

Corollary 1 Let ( u , p ) be Leray’s weak solution defined in a neighborhood of [0 , T ]. If T is an epoch of irregularity, then for every q ≥ 3 and M > 0, we have M � ( u 1 , u 2 )( · , t ) � L q ≥ ( T − t ) (1 − 3 / q ) / 2 and ǫ ( M ) � u 3 ( · , t ) � L q ≥ ( T − t ) (1 − 3 / q ) / 2 , for t < T sufficiently close to T . 15 of 35

Theorem 2 For every M > 0, there exists a constant ǫ ( M ) > 0 with the following property: If ( u , p ) is a suitable weak solution of (NSE) in a neighborhood of Q r ( x 0 , t 0 ) ⊂ D which satisfies 1 � ( | u | 3 + | p | 3 / 2 ) dxdt ≤ M (3) r 2 Q r ( x 0 , t 0 ) and either 1 � | u | 3 dxdt ≤ ǫ (4) r 2 Q r ( x 0 , t 0 ) or 1 � | p | 3 / 2 dxdt ≤ ǫ, (5) r 2 Q r ( x 0 , t 0 ) then u is regular at ( x 0 , t 0 ). 16 of 35

Corollary 2 Let ( u , p ) be Leray’s weak solution defined in a neighborhood of [0 , T ]. If T is an epoch of irregularity, then for every q ≥ 3 and M > 0, we have M � u ( · , t ) � L q ≥ ( T − t ) (1 − 3 / q ) / 2 and ǫ ( M ) � p ( · , t ) � L q / 2 ≥ ( T − t ) (1 − 3 / q ) / 2 , for t < T sufficiently close to T . 17 of 35

Sketch of the proof of Theorem 1. We argue by contradiction. Fix r > 0 and assume that there exists a sequence of suitable weak solutions ( u ( n ) , p ( n ) ), with ( x 0 , t 0 ) a singular point for each of them with 1 � | u ( n ) | 3 + | p ( n ) | 3 / 2 dxdt ≤ M r 2 Q r ( x 0 , t 0 ) and 1 � 3 | 3 dxdt → 0 . | u ( n ) r 2 Q r ( x 0 , t 0 ) 18 of 35

Lemma Let ( u ( n ) , p ( n ) ) be a sequence of suitable weak solutions such that 1 � | u ( n ) | 3 + | p ( n ) | 3 / 2 � � dxdt ≤ M , (6) r 2 Q r ( x 0 , t 0 ) and let 0 < ρ < r . Then there exists a subsequence ( u ( n k ) , p ( n k ) ) such that u ( n k ) → u strongly in L q ( Q ρ ( x 0 , t 0 )) for all 1 ≤ q < 10 / 3 and p ( n k ) ⇀ p weakly in L 3 / 2 ( Q ρ ( x 0 , t 0 )). 19 of 35

Note that the pair ( u , p ) solves the system 2 � ∂ t u i − ∆ u i + u j ∂ j u i + ∂ i p = 0 in D i = 1 , 2 (LS) j =1 ∂ 3 p = 0 in D ∂ 1 u 1 + ∂ 2 u 2 = 0 in D where u ( x 1 , x 2 , x 3 , t ) and p ( x 1 , x 2 , x 3 , t ) are unknown. 20 of 35

Theorem 3 Let ( u , p ) be a solution of ( LS ). Then u is regular. 21 of 35

Theorem 3 implies that we may reduce r and assume M 0 = � u � L 6 ( Q r ( x 0 , t 0 )) < ∞ . Using H¨ older’s inequality we obtain � u � L 3 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 5 / 2 � u � L 6 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 5 / 2 M 0 0 < ρ ≤ r from where 1 ρ 2 / 3 � u � L 3 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 11 / 6 � u � L 6 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 11 / 6 M 0 0 < ρ ≤ r . 22 of 35

Let r 0 ≤ r be sufficiently small so that M 0 ≤ 1 Cr 11 / 6 2 ǫ 1 / 3 W . 0 Then 1 � u � L 3 ( Q r 0 ( x 0 , t 0 )) ≤ 1 2 ǫ 1 / 3 W . r 2 / 3 0 23 of 35

Since u ( n ) → u strongly in L 3 loc ( Q r 0 ( x 0 , t 0 )), we may choose n large enough so that 1 � u ( n ) − u � L 3 ( Q r 0 ( x 0 , t 0 )) ≤ 1 2 ǫ 1 / 3 W . r 2 / 3 0 Thus we obtain 1 � | u ( n ) | 3 dxdt ≤ ǫ W . r 2 Q r ( x 0 , t 0 ) 0 This leads to a contradiction as the above inequality implies that the suitable weak solution u ( n ) is regular at ( x 0 , t 0 ). Note that instead of Wolf’s result we may use Theorem 2. 24 of 35

Sketch of the proof of Theorem 2. First, we prove the assertion under the condition on the pressure. We argue by contradiction. Fix r > 0 and assume that there exists a sequence of suitable weak solutions ( u ( n ) , p ( n ) ), with ( x 0 , t 0 ) a singular point for each of them with 1 � | u ( n ) | 3 dxdt ≤ M r 2 Q r ( x 0 , t 0 ) and 1 � | p ( n ) | 3 / 2 dxdt → 0 . r 2 Q r ( x 0 , t 0 ) 25 of 35

By possibly reducing r > 0 we may assume that u ( n ) → u strongly in L 3 ( Q r ( x 0 , t 0 )). Note that u solves the Burgers-type system 3 � ∂ t u i − ∆ u i + u i ∂ j u j = 0 in D i = 1 , 2 , 3 . j =1 with ∂ 1 u 1 + ∂ 2 u 2 + ∂ 3 u 3 = 0 in D . Therefore u is regular. 26 of 35

Thus, there exists a constant M 0 > 0 (which depends on M ) such that � u � L 6 ( Q r ( x 0 , t 0 )) = M 0 < ∞ . We conclude as in the proof of Theorem 1. Namely, H¨ older’s inequality yields � u � L 3 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 5 / 2 � u � L 6 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 5 / 2 M 0 0 < ρ ≤ r which implies 1 ρ 2 / 3 � u � L 3 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 11 / 6 � u � L 6 ( Q ρ ( x 0 , t 0 )) ≤ C ρ 11 / 6 M 0 0 < ρ ≤ r . 27 of 35